Ž . Journal of Algebra 214, 631635 1999 Article ID jabr.1998.7705, available online at http:www.idealibrary.com on On the Movement of a Permutation Group Peter M. Neumann The Queen’s College, Oxford OX1 4AW, England and Cheryl E. Praeger Department of Mathematics, Uni  ersity of Western Australia, Perth, Western Australia 6907, Australia Communicated by Alexander Lubotzky Received July 23, 1997 1. THE THEOREM  In 1 the second-named author defined the movement ofa permutation Ž . group G,  by the formula g   m  move G  sup       , g  G  4 Ž . and proved that if G has no fixed points and m is finite then n  5m  2,    where n is the degree of G, that is, n   . In 2 it is shown that equality holds if and only if n  3 and G is transitive. In this note we aim to improve the bound: Ž. Ž . THEOREM. 1 If G has no fixed points and move G  m then 1 Ž . n  9 m  3; 2 Ž. 2 equality holds infinitely often; 1 Ž. Ž . Ž . 3 moreo  er , if n  9 m  3 then either n  3 and G  Sym  2 or G is an elementary abelian 3-group and all its orbits ha  e length 3. Ž. Ž. Ž. Part 2 is proved by examples; 1 and 3 will be proved using a sequence of lemmas. 631 0021-869399 $30.00 Copyright  1999 by Academic Press All rights of reproduction in any form reserved.